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Unformatted text preview: e general solution of Eq. (2.1). 106 2 T imeDomain Analysis of ContinuousTime Systems 2 .2 System Response to Internal Conditions: Z eroInput Response
T he z eroinput response yo(t) is t he solution of Eq. (2.1) when the i nput f {t) = 2.2 S ystem Response t o I nternal Conditions: ZeroInput Response Observe t hat t he polynomial Q(A), which is c haracteristic of t he system, has
nothing t o do with t he i nput. For this reason t he polynomial Q(A) is called the
c haracteristic p olynomial of t he system. T he e quation o so t hat Q(A) = 0
Q {D)yo{t) = 0 or
(Dn (2.4a) + an_1D n  l + ... + a iD + ao) yo{t) = 0 (2.4b) A solution t o t his e quation can be obtained systematically. 1 However, we will take
a short c ut by u sing heuristic reasoning. Equation (2.4b) shows t hat a linear combination of y o(t) a nd its n successive derivatives is zero, not a t s ome values of t ,
b ut for all t. S uch a r esult is possible i f a nd o nly i f yo(t) a nd all its n successive
derivatives are o f t he same form. Otherwise their sum can never a dd t o zero for all
values o f t. We know t hat only a n e xponential function e At has this property. So
l et us assume t hat
yo(t) = ce).t
is a solution t o E q. (2.4b). T hen
Dyo{t) d
=  yo = CAe ).t
dt
2 2 D yo{t) = d yo  2  = CA dt e Repeated Roots T he solution of Eq. (2.4) as given in Eq. (2.6) assumes t hat t he n c haracteristic
roots AI, A2, . .. , An a re distinct. I f t here a re repeated roots (same r oot occurring
more t han once), t he form of t he s olution is modified slightly. By direct substitution
we can show t hat t he solution of t he e quation + an_1A n l + ... + alA + ao) eAt = (D  A)2YO(t) = 0 0 is given by For a nontrivial solution o f t his equation,
n
An + an_1A  l (2.7) is called t he c haracteristic e quation of t he s ystem. Equation (2.5c) clearly indicates t hat AI, A2, . .. , An a re t he r oots of t he c haracteristic equation; consequently,
t hey a re called t he c haracteristic r oots of t he s ystem. T he t erms c haracteristic
v alues, e igenvalues, a nd n atural f requencies a re also used for characteristic
roots.:j: T he exponentials e).,t( i = 1 ,2, . .. , n ) in t he z eroinput response are t he
c haracteristic m odes (also known as m odes or n atural m odes) of t he system. T here is a c haracteristic mode for each characteristic root of t he system, a nd
t he z eroinput response is a linear combination o f t he c haracteristic m odes o f t he
system.
T he single most i mportant a ttribute of a n LTIC system is i ts characteristic
modes. Characteristic modes n ot only determine t he z eroinput response b ut also
play a n i mportant role in determining t he z erostate response. In other words, t he
e ntire behavior of a system is d ictated primarily by its characteristic modes. In t he
r est o f t his chapter we shall see the pervasive presence of characteristic modes in
every aspect of system behavior. 2 At DnYo{t) = dnyo = CAne).t
dtn
S ubstituting t hese r esults in Eq. (2.4b), we o btain...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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